1
Genie Chain and Degrees of Freedom of Symmetric MIMO Interference Broadcast Channels
arXiv:1309.6727v1 [cs.IT] 26 Sep 2013
Tingting Liu, Member, IEEE and Chenyang Yang, Senior Member, IEEE
Abstract—In this paper, we study the information theoretic degrees of freedom (DoF) for symmetric multi-input-multi-output interference broadcast channel (MIMO-IBC) with arbitrary configuration. We find the maximal DoF achieved by linear interference alignment (IA), and prove when linear IA can achieve the information theoretic maximal DoF. Specifically, we find that the information theoretic DoF can be divided into two regions according to the ratio of the number of antennas at each base station (BS) to that at each user. In Region I, the sum DoF of the system linearly increases with the number of cells, which can be achieved by asymptotic IA but not by linear IA, where infinite time/frequency extension is necessary. In Region II, the DoF is a piecewise linear function, depending on the number of antennas at each BS or that at each user alternately, which can be achieved by linear IA without the need of infinite time/frequency extension, and the sum DoF cannot exceed the sum number of antennas at each BS and each user. We propose and prove the information theoretic DoF upper-bound for general MIMO-IBC including the system settings in Regions I and II, by constructing a useful and smart genie chain. We prove the achievability of the upper-bound in Region II by proposing a unified way to design closed-form linear IA. From the proof we reveal when proper systems are feasible or infeasible and explain why. The approach of the proof can be extended to more general asymmetric MIMO-IBC. Index Terms—Interference alignment (IA), interference broadcast channel (IBC), degrees of freedom (DoF), genie chain, irresolvable and resolvable ICIs
I. I NTRODUCTION Inter-cell interference (ICI) is a bottle of improving the sum capacity of multi-cell systems, especially when multi-inputmulti-output (MIMO) techniques are used. The degrees of freedom (DoF) can reflect the potential of interference networks, which is the first-order approximation of sum capacity at high signal-to-noise ratio regime [1, 2]. Recently, significant research efforts have been devoted to find the DoF for MIMO interference channel (MIMO-IC) [1–7] and MIMO interference broadcast channel (MIMO-IBC) [9, 10]. For two-cell MIMO-IC with time/frequency varying channels, the information theoretic maximal DoF was found in [2]. For symmetric G-cell M × N MIMO-IC where each base station (BS) and each user have M = N antennas, the authors in [3] shown that the information theoretic maximal DoF per user is M/2, which can be asymptotically achieved by interference alignment (IA) with infinite symbol extensions. This implies that the sum DoF can linearly increase as G, and the interference networks are not interference-limited [3]. Encouraged by such a promising result, many recent works T. Liu and C. Yang are with the School of Electronics and Information Engineering, Beihang University, Beijing 100191, China. (E-mail:
[email protected],
[email protected])
strived to analyze the DoF for MIMO-IC with various setting and devise interference management techniques to achieve the maximal DoF. For symmetric three-cell MIMO-IC, the authors in [1] found the information theoretic maximal DoF, while the authors in [4] found the maximal DoF achieved by linear IA, which shows that these two DoFs are equal. This means that linear IA can achieve the maximal DoF in this case. For symmetric G-cell MIMO-IC where G > M/N (M ≥ N ), the authors in [5–7] derived the information theoretic maximal DoF, which can only be achieved by the asymptotic IA (i.e., with infinite time/frequency extension), where the value of M/N must be an integer in [5] but is no necessary as integer in [6, 7]. In all the other cases, the information theoretic maximal DoF remains unknown [7]. Considering that MIMO-IBC is more complex than MIMOIC and expecting that the results for MIMO-IBC can be extended from MIMO-IC, there are only a few results on MIMOIBC. For symmetric G-cell K-user M × N MIMO-IBC, the information theoretic maximal DoF has been found for the twocell case with two users [8] or with K users each with one antenna [9], and for the G-cell case under special antennas configuration (e.g., N = 1 and M/K is an integer and M/K ≤ G − 1 [10]). For all the other settings, the information theoretic maximal DoF of MIMO-IBC is still unknown. Asymptotic IA requires infinite time/frequency extension, which is not feasible for practical systems. This motivates the study for finding the maximal DoF achieved by linear IA. The maximal DoF can be derived from analyzing the linear IA feasibility. For the MIMO-IC with constant coefficients (i.e., without time/frequency extension), a proper condition was first proposed in [11] by comparing the numbers of equations and variables of the IA condition. When the channels are generic (i.e., drawn from a continuous probability distribution), the authors in [12,13] proved that the proper condition is necessary and the DoF upper-bound of linear IA for symmetric MIMO-IC is (M +N )/(G+1). Moreover, with the finite spatial extension proposed in [1] to avoid the DoF loss due to rounding, it is not hard to show that the upper-bound is achievable for some special cases (i.e., M = N from the results of [12], and either M or N is divisible by the number of data streams per user d from the results of [13]). For the MIMO-IBC with constant coefficients, the proper condition was proved to be necessary in [14, 15] and the DoF upper-bound of linear IA for symmetric MIMO-IBC is (M + N )/(GK + 1). Again considering the finite spatial extension, the upper-bound can be shown to be achievable for some special cases (i.e., G = 2 [16, 17], and symmetric G-cell
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MIMO-IBC where M or N is divisible by d [15]). An alternative approach for deriving the achievable DoF of linear IA is finding the IA transceiver for MIMO-IC [18–20] and MIMO-IBC [21, 22]. Because closed-form IA transceivers are only found for a number of antenna configurations, with this approach it is also difficult to analyze the maximal achievable DoF of linear IA. So far, the following questions are still open for general MIMO-IC and MIMO-IBC: what is the maximal achievable DoF of linear IA? when linear IA can achieve the information theoretic maximal DoF? when the proper systems are feasible? In this paper, we propose and prove the information theoretic maximal DoF and the maximal achievable DoF of linear IA for symmetric G-cell K-user M × N MIMO-IBC, which reduces to general symmetric MIMO-IC with K = 1. We find that the DoF can be divided into two regions according to the ratio of M/N . In Region I, the information theoretic maximal DoF is M N/(M + KN ), which can be achieved by asymptotic IA [6] but not by linear IA. In Region II, the DoF is piecewise linear depending on either M or N , which can be achieved by linear IA. The main contributions are summarized as follows. • By introducing a new way to construct a useful and smart genie chain, we find and proved the information theoretic maximal DoF for symmetric MIMO-IBC with arbitrary configurations. The genie we constructed is independent from the observation of the BS or the user, whose dimension is obtained from the dimension of an irresolvable subspace. Therefore, our genie gain differs from the subspace alignment chain in [1] that cannot be employed to find the DoF upper-bound for general setting. From the proof, we find that there exists another kind of necessary condition for both linear IA and asymptotic IA, which ensures a sort of resolvable ICI (i.e., the irreducible ICIs in [15]) to be eliminated. This answers the question: when proper systems are feasible or infeasible and why. • We find the maximal achievable DoF of linear IA, and prove when linear IA can achieve the information theoretic maximal DoF. This extends the results for three-cell MIMO-IC in [4], whose approach to prove the achievability is not applicable for general setting. By finding the connection of the resolvable ICIs and aligned matrix, we propose a unified way to design the closed-form linear IA for general symmetric MIMO-IBC. When the system configuration falls in Region II, the proposed IA transceiver can achieve the information theoretic maximal DoF. • The basic principle of constructing the genie to find the DoF upper-bound and designing the linear IA transceiver can be extended to more general asymmetric MIMO-IBC and MIMO-IC with arbitrary configurations. The rest of the paper is organized as follows. We describe the system model in Section II. We present the information theoretic maximal DoF and the maximal achievable DoF of linear IA in Section III and prove the results in Sections IV and V, respectively. Conclusions are given in the last section. Notations: Transpose, Hermitian transpose and expectation are represented by (·)T , (·)H , and E{·}, respectively. diag{·} is a block diagonal matrix, I d is an identity matrix of size d. | · | + is the cardinality of a set. (·) is the ceiling operation.
II. S YSTEM M ODEL AND M ETRICS A. System Model Consider a G-cell multiuser MIMO system where G BSs each with M antennas serves K users each with N antennas. Global channel knowledge is assumed available at all BSs. In information theory terminology, this is a symmetric MIMOIBC, denoted as G-cell K-user M × N MIMO-IBC. The received signal of the kth user in the ith cell, user ik , can be expressed as y ik =
G X
H ik ,j xj + nik
(1)
j=1
where x j ∈ CM ×1 is the signal vector transmitted from BS j, H ik ,j ∈ CN ×M is the channel matrix from BS j to user ik whose elements are i.i.d. random variables with a continuous distribution, n ik ∈ CN ×1 is the zero-mean additive white n ik n H Gaussian noise at the user with covariance matrix E{n ik } = 2 σnI N , and i, j ∈ {1, · · · , G}, k ∈ {1, · · · , K}. B. Related Definitions Because we consider IA with or without symbol extension, to avoid confusion, we define several terminologies to be used throughout the paper. Definition 1: For the symmetric MIMO-IBC, the DoF per user is defined as d(M, N ) , lim
γ→∞
R(γ) log γ
(2)
where R(γ) is the achievable rate of each user and γ = 2 xH j x j /(Kσn ) is the signal-to-noise ratio. Definition 2: Linear IA is the IA without any symbol extension or only with finite spatial extension [11]. Definition 3: Asymptotic IA is the IA with infinite time or frequency extension [1]. Definition 4: The DoF per user with finite spatial extension is defined as d(mM, mN ) ¯ d(M, N ) , max (3) m m∈Z+ where m is a finite integer and Z+ is the set of positive integers. ¯ It means that a DoF of md(M, N ) is achievable for each user in the MIMO-IBC when each BS and each user are equipped with mM and mN antennas. With the spatial extension, the DoF per user is no necessary to be an integer, which avoids the loss of DoF due to rounding. For notation simplicity, in the sequel we use d to denote the DoF per user as well as the number of data streams able to be supported for each user by omitting M, N . III. I NFORMATION T HEORETIC M AXIMAL D O F In this section, we first present and compare several DoF bounds for the symmetric MIMO-IBC. Then, we present the information theoretic maximal DoF and show the connection with existing results in the literature.
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In [7], a decomposition DoF bound was introduced, which is the DoF per user achieved by asymptotic IA for G-cell M × N MIMO-IC. It was obtained as M N/(M + N ) by decomposing the antennas at both transmitter and receiver. Using the similar way by decomposing the antennas at the BSs,1 the G-cell K-user M × N MIMO-IBC will reduce to a GK-cell M/K × N MIMO-IC. According to the results for MIMO-IC [7], we can obtain the decomposition DoF bound for the MIMO-IBC as M/K ·N/(M/K +N ) = M N/(M +KN ), which is achievable by asymptotic IA. For the symmetric MIMO-IBC, the decomposition DoF bound is dDecom =
MN M + KN
(4)
The bound is independent of the number of cells, which implies that the sum DoF of the system linearly increases with G. In [11], a proper condition was proposed, which has been proved to be necessary for linear IA in MIMO-IC [12, 13] and in MIMO-IBC [15], when the channels are generic (i.e., drawn from a continuous probability distribution). We call the DoF per user obtained from the proper condition as the proper DoF bound. By counting the number of variables and equations of the IA conditions, the proper condition for MIMO-IBC is M + N ≥ (GK + 1)d [15], which gives rise to the following proper DoF bound. For the symmetric MIMO-IBC, the proper DoF bound is dProper =
M +N GK + 1
(5)
This bound decreases with the number of cells G, so that the sum DoF of the system is bounded by M + N . In the forthcoming analysis, we will find another DoF per user upper-bound, which is a piecewise linear function of M and N . Since when G = 3, K = 1, the DoF upper-bound to be derived reduces to the quantity DoF introduced in [1], we call it as the quantity DoF bound. For the symmetric MIMO-IBC, the quantity DoF bound is ( ) N M A ∀ CnA ≤ M min K+CnA , 1+ K N < Cn−1 Quan CA n−1 d = M N B B min 0, the irresolvable subspace at each BS is nonempty, which is called as the first irresolvable subspace. For a user in the MIMO-IBC, there are (G − 1) interfering BSs each with ((G − 1)KN − M )-dimensional irresolvable subspace. Then, the overall dimension occupied by the irresolvable ICIs seen at the user is (G − 1)((G − 1)KN − M ). Since each user has N -antennas, the remaining + ((G − 1)((G − 1)KN − M ) − N ) -dimensional subspace is irresolvable at the user, which is called as the second irresolvable subspace. Denote SnA as the nth irresolvable subspace, and |SnA | as its dimension. From the above analysis, we have +
|S1A | = ((G − 1)KN − M ) + |S2A | = (G − 1)|S1A | − N + = (G − 1)2 K − 1 N − (G − 1)M
(18a) (18b) (18c)
Again, for each BS, there are (G − 1)K interfering users each with |S2A |-dimensional irresolvable subspace. Then, the overall dimension occupied by the irresolvable ICIs in S2A seen 4 The observation means the desired signals received at a BS, or the ICIs experienced by the BS that is resolvable by the BS.
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by the BS is (G−1)K|S2A |. After reserving resources to remove the M -dimensional resolvable ICIs, each BS still has |S1A |dimensional resource available to further remove ICI. In other words, each BS can resolve at most |S1A |-dimensional irresolvable ICIs. Therefore, the dimension of the third irresolvable subspace is + |S3A | = (G − 1)K|S2A | − |S1A | (19) We can proceed to compute the irresolvable subspace at each user again, which is the fourth irresolvable subspace if nonempty, and so on. These irresolvable subspaces constitute an irresolvable subspace chain, called subspace chain for short. Following similar analysis, the dimension of the nth (n ≥ 3) irresolvable subspace can be obtained as ¯ n−1 |S A | − |S A | + |SnA | = (G − 1)K (20) n−1 n−2 To express the dimension in a unified way, we denote A |S−1 | = M,
|S0A | = N
(21)
Then, (20) holds ∀n ∈ Z+ . When one irresolvable subspace in the chain is empty, the subspace chain stops. Specifically, if the dimension of the irresolvable subspace satisfies > 0 ∀m < n A |Sm | (22) = 0 ∀m = n the length of the subspace chain is n, where the chain contains n − 1 nonempty irresolvable subspaces and one empty irresolvable subspace at the end of the subspace chain. To understand the subspace chain, in the following we provide an example (Ex 1) with different values of n in Fig. 3, where G = 3, K = 2. Ex 1(a): For the case shown in Fig. 3(a) where M ≥ 4N , from (20) we have |S1A |
+
+
= ((G − 1)KN − M ) = (4N − M ) = 0
(23)
The subspace chain stops and the length of subspace chain is 1. All the ICIs are resolvable for the BS so that they cannot be aligned. In other words, the users need not to and cannot help the BS to remove the ICI. Ex 1(b): For the case shown in Fig. 3(b) where 4N > M ≥ 7/2N , from (20) we have +
+
|S1A | = ((G − 1)KN − M ) = (4N − M ) > 0 + + |S2A | = (G − 1)|S1A | − N = (7N − 2M ) = 0
(24a) (24b)
Since |S1A | > 0, the subspace chain continues. In this case, it is possible for the users to align some ICIs into the first irresolvable subspace, i.e., they can help the BS to eliminate part of ICIs. Since |S2A | = 0, the length of subspace chain is 2. As a result, it is impossible to further align the irresolvable ICIs in S1A into the second irresolvable subspace. Ex 1(c): For the case shown in Fig. 3(c) where 7/2N > M ≥ 24/7N , from (20) we have +
+
|S1A | = ((G − 1)KN − M ) = (4N − M ) > 0 (25a) + + A A |S2 | = (G − 1)|S1 | − N = (7N − 2M ) > 0 (25b) + A A A + |S3 | = (G − 1)K|S2 | − |S1 | = (24N − 7M ) = 0 (25c)
Since |S1A | > 0, |S2A | > 0, it is possible to further align the irresolvable ICIs into S2A , i.e., the users can help the BS to eliminate more ICIs than the case shown in Fig. 3(b). Since |S3A | = 0, the length of subspace chain is 3. In [1], the authors introduced a subspace alignment chain, which is comprise of the signal subspace where two users are aligned. The subspace alignment chain is only appropriate for analyzing the case with only two interfering BSs or users, i.e., G = 3, K = 1, but cannot be extended into general cases. Our irresolvable subspace chain is different from the subspace alignment chain in [1], which is applicable for general symmetric MIMO-IBC. B. Genie Chain and DoF Upper-bound Since the irresolvable and resolvable ICIs are mutually independent, the linear combination of irresolvable ICIs can serve as the genie required in Lemma 2. This suggests that we can determine the dimension of the genie5 from the dimension of the irresolvable subspace. To illustrate how to determine the genie dimension and then the DoF upper-bound, we again investigate previous example. Ex 1(a): For the case shown in Fig. 3(a), there is no irresolvable subspace as indicated in (23), such that the BS does not need any genie to resolve all ICIs, i.e., |G1A | = 0. Upon substituting into (17), we obtain a DoF per user upper-bound as d≤
M K + (G − 1)K
(26)
To ensure that each user has enough dimension to resolve the desired signals, the DoF per user also needs to satisfy d≤N
(27)
From (7a), we have C1A = (G − 1)K and C0A = ∞. Therefore, from (26) and (27) we can obtain the DoF per user upper-bound as ( ) M N d ≤ min , (28) K + C1A 1 + CKA 0
Ex 1(b): For the case shown in Fig. 3(b), from (24) we have |S1A | > 0 and |S2A | = 0. Therefore, we can introduce a genie (denoted by G1A ) to help each BS to resolve the ICIs, but cannot introduce a genie (denoted by G2A ) to help each user to resolve G1A . Further considering that genie G1A should lie in subspace S1A , we have (G − 1)Kd ≤ (M − Kd) + |G1A | (G −
1)|G1A | |G1A |
≤d ≤
(29a) (29b)
|S1A |
(29c)
From (29b), we have |G1A | ≤ d/(G − 1). Upon substituting into (29a), we obtain a DoF per user upper-bound as d≤
M K + (G − 1)K −
1 G−1
(30)
5 It is shown from Lemma 2 that to derive the DoF upper-bound, we only need to obtain the dimension of the genie and no need to obtain the expression of the genie. Therefore, unlike [7], we do not show the expression of the genie.
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MSs
BSs
MSs
BSs
MSs
MSs
BSs
A 1
MSs
BSs
A 1 A 2
Cell 1 A 1
A 1
Cell 2
A 2
A 1
A 1
A 1 A 2
Cell 3
A 2
A 1
A 1
0
A 1
(a) M≥4N (n=1)
0
A 2
(b) 4N>M≥7/2N (n=2)
0
A 1
0
A 2
0
A 3
0
(c) 7/2N>M≥24/7N (n=3)
Fig. 3. Example of irresolvable subspace chain, G = 3, K = 2.
Substituting (29c) and |S1A | = (G − 1)KN − M into (29a), we obtain d≤
(G − 1)KN N = K K + (G − 1)K 1 + (G−1)K
(31)
From (7a), we have C2A = (G − 1)K − 1/(G − 1) and C1A = (G − 1)K. Then, from (30) and (31) the DoF per user upperbound can be expressed as ) ( N M , (32) d ≤ min K + C2A 1 + CKA 1
Ex 1(c): For the case shown in Fig. 3(c), from (25) we have |S1A | > 0, |S2A | > 0, and |S3A | = 0. Therefore, we can introduce a genie G1A to help each BS to resolve the ICIs, a genie G2A to help each user to resolve G1A , but cannot introduce a genie G3A to help the BS to resolve G2A . Further considering that genie G2A lies in subspace S2A , we have (G − 1)Kd ≤ (M − Kd) + |G1A | (G − 1)|G1A | (G − 1)K|G2A | |G2A |
≤d+ ≤ ≤
|G2A |
(33a) (33b)
|G1A | |S2A |
(33c) (33d)
From (33c), we obtain |G2A | ≤ |G1A |/((G ing into (33b), we have |G1A | ≤ d/((G − 1)
− 1)K). Substitut− 1/((G − 1)K)).
Then, substituting into (33c), we obtain d≤
M K + (G − 1)K −
1
(34)
1 G−1− (G−1)K
Substituting (33d) and |S2A | = (G − 1)|S1A | − N into (33b), we have |G1A | ≤ |S1A | − (N − d)/(G − 1). Upon substituting into (33a) and considering |S1A | = (G − 1)KN − M , we have d≤
N 1+
K 1 (G−1)K− G−1
(35)
From (7a), we have C3A = (G − 1)K − 1/((G − 1) + 1/((G − 1)K)) and C2A = (G − 1)K − 1/(G − 1). Further considering (34) and (35), the DoF per user upper-bound is ( ) M N d ≤ min (36) , K + C3A 1 + CKA 2
Now we can summarize the approach we construct the genie. When each BS cannot resolve ICIs, we construct genie G1A in the first irresolvable subspace S1A to help the BS. When G1A is not irresolvable in S1A , we construct G2A in S2A to help G1A . In this way, the genie will not provide any redundant information, which simplifies the derivation of the DoF upper-bound and the upper-bound is tight. C. Proof For Region II-A From the above example we can find that the DoF upperbound can be obtained by finding the maximal dimension of the genie, which depends on the length of genie chain. Therefore, to prove the result for general MIMO-IBC, we only need to find the length of genie chain for different antenna configurations. Using mathematical induction, we can prove that + |SnA | = qnA N − pA (37) nM From (B.2) in Appendix B we know that CnA = qnA /pA n . Then, the dimension of the irresolvable subspace can be rewritten as + M A |SnA | = pA N C − (38) n n N Since {CnA } is a monotonic decreasing sequence, we know A , (22) holds, i.e., from (38) that when CnA ≤ M/N < Cn−1 the length of the subspace chain is n. Therefore, Region II-A defined in (9) can be divided into multiple regions, where the nth region is A RII−A , [CnA , Cn−1 ), ∀n ∈ Z+ n
(39)
A When M/N ∈ RnII−A , from (22) we have |Sm | > 0, ∀m < A n. Hence we can introduce the genie G1 to help each BS to
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resolve ICIs, the genie G2A to help each user to resolve G1A , and A A . This means so on until introduce a genie Gn−1 to resolve Gn−2 (G − 1)Kd ≤ (M − Kd) + |G1A | (G − 1)|G1A | (G − 1)K|G2A |
≤d+ ≤ .. .
|G2A |
|G1A |
+
(40a) (40b)
|G3A |
(40c)
A A A ¯ n−2 |Gn−2 (G − 1)K | ≤ |Gn−3 | + |Gn−1 |
(40d)
To express these inequalities in a unified way, we denote A |G−1 | = M − Kd
|G0A |
(41a)
=d
(41b)
¯ n in (16) is a sequence whose value is K Considering that K and 1 alternately, we can rewrite the inequalities in (40) as (G− ¯ m−1 |G A | ≤ |G A | + |G A |, ∀m < n. 1)K m m−1 m−2 Moreover, from (22) we have |SnA | = 0. Consequently, we A A and genie Gn−1 cannot introduce a genie GnA to resolve Gn−1 A A A ¯ is in subspace Sn−1 , i.e., (G − 1)Kn−1 |Gn−1 | ≤ |Gn−2 | and A A |Gn−1 | ≤ |Sn−1 |. Combining these inequalities, we obtain ¯ m−1 |G A | ≤ |G A | + |G A |, (G − 1)K m−1 m−2 m A A ¯ m−1 |Gm−1 (G − 1)K | ≤ |Gm−2 |, A |Gm−1 |
≤
A |Sm−1 |,
UH ik H ik ,j V j = 0 , ∀i 6= j
(45b)
where V j and U ik denote the transmit and receive matrices of BS j and user ik . (45a) is a rank constraint to convey the desired signals for each BS and each user, which is equivalent V j ) = KdQuan and rank(U U ik ) = dQuan , and (45b) is to rank(V the zero-forcing constraint to eliminate ICI. In [1, 4], the authors proved that the quantity DoF bound is achievable for three-cell MIMO-IC by finding a way to design the feasible linear IA. To understand what is new in designing the feasible IA to achieve the DoF upper-bound for general MIMO-IBC, we briefly review the method in [1, 4] through an example where G = 3, K = 1, M/N = 5/7 (Ex 2). A. Example of Feasible Linear IA
∀m < n (42a)
From (6), the quantity DoF bound for Ex 2 is dQuan = 3N/7 = 3M/5, i.e., N = 7/3dQuan , M = 5/3dQuan . In [1, 4], the transmit matrices of three BSs are designed as
∀m = n (42b)
¯ (l) }, l = 1, 2, 3 W V (l) ∈ null{H
∀m = n (42c)
In Appendix D, we show that the dimension of the mth (m ≤ n) genie satisfies ¯ m−1 |G A | K m−1 A Cn−m ¯ m−1 |S A | − |G A | K m−1 m−1 A A |Gm | ≤ |Sm | − A Cn−m−1
A |Gm |≤
To support d = dQuan data streams for each user, the IA conditions for MIMO-IBC is [15] H 0 U i1 H i1 ,i .. Quan .. (45a) rank . V i = Kd . H H iK ,i 0 U iK
(43a) (43b)
Since |G0A | = d from (41b), we can obtain the DoF upperbound from (43a) and (43b) with m = 0, respectively. Then, substituting (21) and (41) into (43a) and (43b), we have d ≤ A . From which we can (M − Kd)/CnA and d ≤ N − Kd/Cn−1 obtain the DoF per user upper-bound as follows, M N d ≤ min (44) , K + CnA 1 + K CA n−1
This completes the proof for Theorem 1. V. P ROOF OF T HEOREM 2 We have known from Section III.A that the decomposition DoF bound is achievable by asymptotic IA in [18] with arbitrary antenna configuration. In the sequel, we only need to prove that the quantity DoF bound is achievable by linear IA for the MIMO-IBC in Region II. The basic idea of the proof is to find a linear IA transceiver that can support d = dQuan data streams for each user, which is borrowed from [1, 4]. To prove that the linear IA is feasible, we need to prove that it satisfies the IA conditions.
(46)
A} denotes the null space of A , where null{A ¯ (1) = H 11 ,2 H 11 ,3 H ∈ C2N ×3M H 21 ,3 H 21 ,1 H 21 ,3 H 21 ,1 ¯ H (2) = ∈ C2N ×3M H 31 ,1 H 31 ,2 ¯ (3) = H 31 ,1 H 31 ,2 H ∈ C2N ×3M H 11 ,2 H 11 ,3 In fact, W V (l) is a rearranged transmit matrix comprising of parts of transmit matrices of different BSs. By expressing the V j(1) , V j(2) , V j(3) ], transmit matrices of BS j in (45) as V j = [V W V (l) can be expressed as h iT dQuan W V (1) = V T2(1) , V T3(1) , V T1(1) ∈ C3M × 3 h iT dQuan W V (2) = V T3(2) , V T1(2) , V T2(2) ∈ C3M × 3 iT h dQuan W V (3) = V T1(3) , V T2(3) , V T3(3) ∈ C3M × 3 (46) can also be written as the following aligned equations H 11 ,2V 2(1) + H 11 ,3V 3(1) = 0 H 21 ,3V 3(1) + H 21 ,1V 1(1) = 0 H 21 ,3V 3(2) + H 21 ,1V 1(2) = 0 H 31 ,1V 1(2) + H 31 ,2V 2(2) = 0 H 31 ,1V 1(3) + H 31 ,2V 2(3) = 0 H 11 ,2V 2(3) + H 11 ,3V 3(3) = 0
(47a) (47b) (47c) (47d) (47e) (47f)
¯ (l) , l = 1, 2, 3 in (46) we can see which ICIs Then, from H are aligned together. Hence, these matrices are called aligned matrices in the following.
9
In [1, 4], the receive matrices of the users are designed as QU U ik ∈ null{Q ik }, i = 1, 2, 3, k = 1
(48)
where Q U ik contains all the ICIs observed by user ik , which is H 11 ,2V 2 , H 11 ,3V 3 ] ∈ C2d QU 11 = [H
Quan
×N
Quan
×N
Quan
×N
H 21 ,1V 1 , H 21 ,3V 3 ] ∈ C2d QU 21 = [H H 31 ,1V 1 , H 31 ,2V 2 ] ∈ C2d QU 31 = [H
To show the feasibility of the designed linear IA, it needs to check whether the transceiver in (46) and (48) satisfy the IA conditions in (45). From the analysis in [4], we know that ¯ (l) ) = 2N, l = 1, 2, 3. Since H ¯ (l) has 3M columns, rank(H there exists 3M − 2N = N/7 = 1/3dQuan -dimensional null ¯ (l) . Since H ¯ (l) , l = 1, 2, 3 are independent with space of H each other, their null space is also independent, i.e., the transmit matrices W V (l) , l = 1, 2, 3 in (46) are mutually independent. V j ) = 3(3M − 2N ) = dQuan . Therefore, we have rank(V Moreover, from the aligned equations in (47) we see that there are two equations for each user (e.g., (47b) and (47c) for user 21 ). This indicates that the designed transmit matrices can help each user to eliminate 2(3M − 2N ) = 2/3dQuan ICIs. Consequently, each user only suffers from ((G − 1)K − 2/3)dQuan = Quan QU 4/3dQuan ICIs, i.e., rank(Q . Since the dimenik ) = 4/3d U QU sion of the null space of Q ik is N − rank(Q ik ) = (7/3 − Quan Quan 4/3)d = d , the user can eliminate all the remaining ICIs and have enough space to receive the desired signals, i.e., U ik ) = dQuan . This UH ik H ik ,j V j = 0 , ∀i 6= j and rank(U indicates that the transceiver meets the IA conditions. From this example, we see that for designing the linear IA transceiver, we need to find the align matrices, with which the transmit and receive matrices can be easily obtained. Since the align matrices reflect which ICIs must be and can be aligned, which depends on the processing abilities of both the BSs and users, they are very hard to construct to ensure the resulting transceiver feasible. This suggests that one key step to prove linear IA feasibility is to construct the aligned matrices. Another key step is to check whether the designed transceiver satisfies the IA conditions. These two steps cannot be extended from those for three-cell MIMO-IC in [1,4] and are difficult for general MIMO-IBC, as will be shown soon. In the following, we start by investigating the aligned matrices for MIMO-IBC to prove the theorem. Then, we design the linear IA transceiver and check whether it satisfies the IA conditions. Again, since the proof for Region II-B is similar to that for Region II-A, we only present the proof in Region II-A for conciseness.
showed that when the length of genie chain is n, one aligned matrix becomes H 11 ,2 H 11 ,3 H 21 ,3 H 21 ,1 ¯ (1) = H ∈ CnN ×(n+1)M H H 3 ,1 3 ,2 1 1 .. .. . . and other aligned matrices have the same structure (therefore, ¯ (1) as an example). Since the aligned matrices with we take H different values of n have identical structure, they can be constructed in a unified way. Unfortunately, except for this special case the generalized Fibonacci sequence is not the arithmetic sequence any more such that the aligned matrices with different n do not have similar structure. For example, in the case of G = 4 and K = 1, when n = 1, one aligned matrix is ¯ (1) = H 1 ,2 H 1 ,3 H 1 ,4 H 1 1 1 and when n = 2, the aligned matrix becomes H 11 ,2 H 11 ,3 H 11 ,4 ¯ (1) = H 21 ,4 H 21 ,1 H 21 ,3 H H 31 ,1 H 31 ,2 H 31 ,4 H 31 ,1 As a result, it is very hard to construct the aligned matrices in a unified way as in [4], if not impossible. Instead of constructing the aligned matrices for general MIMO-IBC directly as in [4], we first show their existence and their dimension by finding a connection of the aligned matrix and resolvable ICIs. In fact, when the length of the genie chain is given, the structure of the aligned matrices is given. H ik ,j ” in each aligned matrix Specifically, all the sub-matrices “H is determined. When the dimension of the aligned matrix is known, i.e., the number of sub-matrices is known, we can find the full rank aligned matrix simply by multiplying a coefficient of “0” or “1” with each sub-matrix. Based on the analysis in section IV.A, we know that all the ICIs can be divided into resolvable ICIs and irresolvable ICIs. For the considered MIMO-IBC, (17) can be rewritten as M≥
Quan Quan − |G A | (49) Kd | {z } + |(G − 1)Kd {z } |{z} Desired signal Total ICI Irresolvable ICIs {z } | Resolvable ICIs
It indicates that to ensure linear IA to be feasible, each BS should have enough resources to convey the desired signals and eliminate the resolvable ICIs. From (13b) and (6), we know that when CnA ≤ M/N ≤ A Cn−1 , the necessary conditions of IA feasibility are M≥
B. Aligned Matrix In [4], the authors find a way to construct the aligned matrix for three-cell MIMO-IC, which is however very difficult to extend to general cases. This is because the size of the aligned matrix is related to the generalized Fibonacci sequence in (15), which will be explained soon. Only when G = 3, K = 1, from B A B (15) we have pA n = qn = n and qn = pn = n + 1, which reduces to the arithmetic sequence. In this case, [4, Lemma 2]
N≥
Quan A Quan Kd d | {z } + C | n {z }
Desired signal
Resolvable ICIs
Quan d | {z }
A + K/Cn−1 dQuan | {z }
Desired signal
(50a) (50b)
Resolvable ICIs
By comparing with the necessary condition in (49), this implies that there will exist CnA dQuan resolvable ICIs for each BS and A K/Cn−1 dQuan resolvable ICIs for each user. The resolvable ICIs can be expressed with the aligned matrix if the number of interfering users or BSs is an integer, e.g., for
10
the cases in in [4]. To do this for general MIMO-IBC, we need A to rewrite CnA and K/Cn−1 in (50a) and (50b) in other forms, which may not be integers. Considering CnA = qnA /pA n , (50a) and (50b) become A Quan pA + qnA dQuan n M ≥ pn Kd A qn−1 N
≥
A qn−1 dQuan
+
Quan pA n−1 Kd
(51a) (51b)
which can be interpreted as that the ICIs from qnA users are reA solvable for pA n BSs and the ICIs from pn−1 BSs are resolvable A for qn−1 users. In the sequel, we illustrate how to express the resolvable ICIs by the aligned matrix. For the example in Section V.A (i.e., Ex. 2), we know from [4] that one necessary condition of IA feasibility is 3M ≥
Quan 3d | {z }
Desired signal
+
Quan |2d{z }
(52)
Resolvable ICIs
This implies that the ICIs from two users to three BSs are ¯ (1) in (46) as the resolvable ICIs. Take the aligned matrix H an example, which is related to three BSs (BSs 2, 3, and 1) U 11 , U 21 }, and two users (users 11 and 21 ). Let U¯ = diag{U H ¯ ¯ ¯ ¯ ¯ ¯ V = diag{V 2 , V 3 , V 1 }. Then, U H (1) represents the ICIs from users 11 and 21 to BSs 2, 3, and 1, which is expressed by ¯ (1) . Since rank H ¯ (1) = 2N ≤ 3M [4], the aligned matrix H we have H ¯ ¯ (53) rank U¯ H (1) = rank(U ) which implies that the ICIs from user 11 and user 21 are not aligned and can be resolvable for the three BSs. It suggests that these ICIs must be and can be eliminated by the BSs. When we design the transmit matrices as in (46), the BSs can eliminate all resolvable ICIs and then the users can eliminate the remaining ICIs. In this way, the designed linear IA is feasible to achieve d = dQuan for each user. This example indicates that the aligned matrices exist if we can find the resolvable ICIs, and then the DoF upper-bound is achievable. In general cases of symmetric MIMO-IBC, from (50a) and (50b) we know that the resolvable ICIs exist. Specifically, when A CnA ≤ M/N ≤ Cn−1 , from (51a) and (51b) we know that A A the ICIs from qn users to pA n BSs and those from pn−1 BSs A to qn−1 users are resolvable. Since the resolvable ICIs can be expressed by aligned matrices, there exist one class of full A rank aligned matrices of size qnA N × pA n M with rank qn N (as implied by (53)) as well as another class of aligned matrices of A A size qn−1 N × pA n−1 M with rank pn−1 M . To distinguish these ¯ V and two classes of aligned matrices, we denote them as H (l) ¯ U , respectively. H (l) The aligned matrices are not unique. For the case in Ex. 2, there are three aligned matrices. In fact, the number of aligned matrices depends on the number of genie chains. For Ex 1(a) shown in Fig. 3(a), we only show one genie chain that stops at BS 3. Besides this genie chain, there are two other chains that stop at BSs 1 and 2, respectively. Consequently, we can obtain three different aligned matrices from the three genie chains. For Ex 1(b) in Fig. 3(b), we can obtain six different genie chains that stop at users 11 , 12 , 21 , 22 , 31 , and 32 , respectively. Correspondingly, there are six align matrices. Since the genie
chain stops at G BSs or GK users alternately, the number of aligned matrices are also G or GK alternately. In general cases, A ¯ n aligned matrices when CnA ≤ M/N < Cn−1 , there are GK V U ¯ and GK ¯ . ¯ n−1 aligned matrices H H (l) (l) The above analysis yields the following observation. Observation 1 (Existence of Aligned Matrices): For the A symmetric MIMO-IBC, when CnA ≤ M/N < Cn−1 , there ¯ V ∈ CqnA N ×pAn M satisfying ¯ n aligned matrices H exist GK (l) ¯ V = qA N (54) rank H n (l) U
A
A
qn−1 N ×pn−1 M ¯ ¯ n−1 aligned matrices H and GK satisfy(l) ∈ C ing ¯ U = pA M rank H (55) (l) n−1
The dimension of the aligned matrices reflects how many ICIs should be aligned, but does not indicate which ICIs should be aligned (fortunately this does not matter). In fact, given the dimension and structure of the aligned matrices, we can find the expressions of the aligned matrices by exhaustive searching, which is of low complexity. C. Linear IA Design Once the aligned matrices are found, the linear IA transceiver can be obtained as follows, whose expression depends on the value of M/N . A A • When Dn−1 ≤ M/N < Cn−1 , where the feasible region is limited by the transmit antenna at each BS M (the vertical boundary of each triangle region in Fig. 2), the transmit and receive matrices are designed as follows V
¯ } W V (l) ∈ null{H (l) U ik ∈
•
QU null{Q ik }
(56a) (56b)
U U U QU where Q U ik = [Q ik ,1 , · · · , Q ik ,i−1 , Q ik ,i+1 , · · · , Q ik ,G ] ∈ Quan ×N C(G−1)Kd contains all the ICIs observed by user U H ik , Q ik ,j = V H j H ik ,j denotes the ICIs observed by user ik from BS j, and W V (l) is the rearranged transmit matrix that is comprised of parts of the transmit matrices from different BSs. A When CnA ≤ M/N ≤ Dn−1 , where the feasible region is limited by the receive antenna at each user N (the horizon boundary of each triangle region in Fig. 2), the transmit and receive matrices are designed as follows U
¯ )H } W U (l) ∈ null{(H (l) Vj ∈
QVj null{Q
}
(57a) (57b)
QVj,1 , · · · , Q Vj,j−1 , Q Vj,j+1 , · · · , Q Vj,G ] ∈ where Q Vj = [Q Quan ×M C(G−1)Kd contains all the ICIs observed by BS V H UH j, Q j,i = [U i1 H j,i1 , · · · , U iK H j,iK ] denotes the ICIs observed by BS j from all the users in cell i, and W U (l) is the rearranged receive matrix that is comprised of parts of the receive matrices from different users. To prove the quantity DoF bound as achievable, we need to check whether the designed IA transceiver meets the IA conditions in (45).
11
D. Check of IA Feasibility From (56b) and (57b), we know that the receive matrix of each user and the transmit matrix of each BS are in the null space of their respective ICIs, hence, we have U H ik H ik ,j V j = 0 , ∀i 6= j. That is to say, the transceiver satisfies (45b). In the following, we check whether the transceiver satisfies (45a), again depending on the value of M/N . A A 1) Dn−1 ≤ M/N < Cn−1 : Since W V (l) is a rearranged matrix from V j , j = 1, · · · , G, the number of transmit vectors in all of W V (l) is the same as that in all of V j , j = 1, · · · , G, which is GKd. ¯ n aligned From Observation 1, we know that there are GK V ¯ , l = 1, · · · , GK ¯ n , which correspond to GK ¯n matrices H (l) ¯ n. rearranged transmit matrices W V (l) , l = 1, · · · , GK ¯ n) = Therefore, each matrix W V (l) contains GKd/(GK V Quan ¯ ¯ Kd /Kn transmit vectors. Considering that H (l) has pA nM columns, W V (l) has pA M rows. Consequently, each coln umn contains pA transmit vectors. Therefore, W V (l) has n ¯ n pA ) columns. KdQuan /(K n V j ) = KdQuan (i.e., the designed transTo ensure rank(V mit vectors are independent), it is necessary to ensure the W V (l) ) = row vectors of W V (l) independent, i.e., rank(W ¯ n pA ). Considering (56a), this means that the folKdQuan /(K n lowing condition should be satisfied Quan ¯ V } ≥ Kd (58) dim null{H (l) ¯ n pA K n ¯ V has pA M columns, from the rank nullity theorem Since H (l) n ¯V . ¯ V } = pA M − rank H [26] we obtain dim null{H n (l) (l) Considering (54), we have ¯ V } = pA M − q A N dim null{H (59a) (l) n n Quan K A ≥pA dQuan (59b) − qnA 1 + A n K + Cn d Cn−1 K K =qnA +1− +1 dQuan (59c) A A Cn Cn−1 Quan A =qnA CnA − Cn+1 d (59d) qnA dQuan A pA n pn+1 KdQuan = ¯ A Kn pn =
(59e) (59f)
A where (59b) comes from M/N ≥ Dn−1 = A A A K + Cn / 1 + K/Cn−1 and N ≥ (1 + K/Cn−1 )dQuan in (6), (59c) follows from CnA = qnA /pA n in (B.2), (59d) is due to A A K/CnA = (G − 1)K − Cn+1 and K/Cn−1 = (G − 1)K − CnA A A A in (7a), (59e) comes from Cn − Cn+1 = 1/(pA n pn+1 ) in A A ¯ n, (B.4a), and (59f) comes from the fact that qn /pn+1 = K/K which can be proved using the mathematical induction. Therefore, (58) is satisfied. U ik ) = dQuan , from (56b) we know that it To ensure rank (U is necessary to satisfy Quan QU dim null{Q (60) ik } ≥ d QU QU To obtain dim null{Q ik } , we first need to find rank(Q ik ). V A Quan ¯ ¯ Since H (l) has qn N rows and W V (l) has Kd /(Kn pA n)
¯ n pA columns, from (56a) we can obtain qnA · KdQuan /(K n) = A Quan ¯ /Kn aligned equations (as exemplified in (47)). Cn Kd ¯ n pA ) ICIs These aligned equations can eliminate KdQuan /(K n ¯ for all users. Further considering that there are GKn rearranged transmit matrices and GK users, all the aligned equations can eliminate GKn CnA KdQuan · = CnA dQuan ¯n GK K ICIs for each user. As a result, each user only suffers from ((G − 1)K − CnA )dQuan ICIs, then from the definition in (56) we have A Quan A QU rank(Q . Since K/Cn−1 = (G − ik ) = ((G − 1)K − Cn )d A U A Qik ) = K/Cn−1 1)K − Cn from (7a), we have rank(Q dQuan . U Since Q ik has N columns, from the rank nullity theorem [26] and (6), we have K Quan QU d ≥ dQuan (61) dim null{Q ik } = N − A Cn−1 Therefore, (60) satisfies. Consequently, the designed transmit and receive matrices satisfy (45a). A : Similarly, we can prove that 2) CnA ≤ M/N ≤ Dn−1 the designed transmit and receive matrices satisfy (45a) in this case. As a result, the quantity DoF bound in Region II-A can be achieved by the linear IA. This completes the proof for Theorem 2. VI. C ONCLUSION In this paper, we found the information theoretic maximal DoF for the G-cell K-user M × N symmetric MIMO-IBC. We answered the questions for general symmetric MIMO-IC and MIMO-IBC regarding what the maximal achievable DoF of linear IA is, when linear IA can achieve the information theoretic maximal DoF and when the proper systems are feasible. According to the ratio of M/N , the DoF can be divided into two regions. In Region I, the information theoretic maximal DoF per user is M N/(M + KN ), which can be achieved by asymptotic IA. In this region, the sum DoF of the system can linearly increase with G but at the cost of infinite symbol extension. In Region II, the information theoretic maximal DoF is a piecewise linear function, which is achievable by linear IA. In this region, the sum DoF is bounded by M + N . From the proof of the main results, we revealed when a proper system is feasible and when is not. As a by-product of finding the connection of the resolvable interference and aligned matrix, we proposed a unified way to design the closed-form transceiver for linear IA, which can achieve the information theoretic maximal DoF in Region II. The basic idea of the proof can be extended to more general asymmetric MIMO-IBC. A PPENDIX A P ROOF OF L EMMA 1 First, we compare the decomposition DoF bound and the proper DoF bound. From (4) and (5), we have MN M +N dProper − dDecom = − GK + 1 M + KN M 2 − (G − 1)KM + KN 2 = (A.1) (GK + 1)(M + KN )
12
From (7a) and (7b), it is not difficult to express the numerator A of (A.1) as M 2 − (G − 1)KM + KN 2 = (M − C∞ N )(M − B B A C∞ N ). Therefore, when C∞ < M/N < C∞ (i.e., in Region I), we have dProper − dDecom < 0. By contrast, when M/N ≥ A B C∞ or M/N ≤ C∞ , (i.e., in Region II), we have dProper − Decom d ≥ 0. Second, we compare the decomposition DoF bound and the quantity DoF bound in Region II. Defining ρ , M/N , then (4) can be rewritten as dDecom = CnA
N M = K +ρ 1 + K/ρ A Cn−1 ,
(A.2) CnA
When ≤ M/N < we have ρ ≥ and 1/ρ > A 1/Cn−1 . Substituting into (A.2), we have M N , (A.3) dDecom ≤ min K + CnA 1 + K CA
According to the relationship between the generalized continue fraction and generalized Fibonacci sequence-pairs in [25], we can rewrite Cnα , ∀n ∈ Z+ , α = A, B as Cnα =
Proper
d
≥
K + 1 + CnA +
K A Cn−1
GK + 1
dQuan = dQuan
(A.4)
A When M/N = Dn−1 , from (6) we have M = (K + CnA )dQuan A and N = (1 + K/Cn−1 )dQuan . Substituting into (5), we have dProper = dQuan . B < M/N ≤ CnB , we can obtain Similarly, when Cn−1 Proper Quan B d ≥d and the equality holds when M/N = Dn−1 . Consequently, we obtain (12b).
A PPENDIX B P ROOF OF C OROLLARY 1
n
According to the properties of continue fraction [25, Theorem 2.1], Cnα , ∀n ∈ Z+ can be expressed as CnA = (G − 1)K − CnB =
CnA = (G − 1)K − CnB =
1 G−1−
1 1 (G−1)K− ···
1 G−1−
1 (G−1)K−
(B.1a) (B.1b)
1 G−1− 1 ···
n X
n−1 X
1
pA pA m=1 m m+1
1
pA pA m=1 m m+1
(B.4a) (B.4b)
+ A Since pA m ≥ 0, ∀m ∈ Z , we know that {Cn } is a monoB tonically decreasing sequence, whereas {Cn } is monotonically increasing sequence. According the monotonicity of {Cnα }, after some regular manipulations, (B.3) (i.e., (14)) can be equivalently derived as ( M ≥ K + CnA d A ∀ CnA ≤ M/N < Cn−1 (B.5a) N ≥ 1 + CK d A n−1 ( B M ≥ K + Cn−1 d B B ∀ Cn−1 ≤ M/N ≤ Cn−1 (B.5b) N ≥ 1 + CKB d n
Comparing with (6), we know that from (B.5) we can obtain d ≤ dQuan , and from d ≤ dQuan we can obtain (B.5), i.e., (14). Since Theorem 1 indicates that the quantity DoF bound dQuan is the DoF upper-bound, this means that (14) is the necessary condition for the IA feasibility. Since Theorem 2 indicates that the quantity DoF bound is achievable, therefore (14) is also the sufficient condition. Further from (B.5) we see that the conclusion holds true in Region II. A PPENDIX C P ROOF OF L EMMA 2 Let W denote the messages transmitted from all BSs, Y m j m and GA denote the observation and genie for BS j. Following the similar analysis as in [7, 24], we have m
From (14), we know that if pM ≥ qN , we have pM ≥ (pK + q)d. Otherwise, N ≥ (pK + q)d/q, ∀(p, q) ∈ A ∪ B. From the recursive formula in (7), we can express Cnα , ∀n ∈ + Z as a form of generalized continued fraction [25], i.e.,
(B.2)
α where pα n and qn are defined in (15). Therefore, (14) can be rewritten as ( M ≥ (K + Cnα)d, ∀M/N ≥ Cnα (B.3) N ≥ 1 + CKα d, otherwise.
n−1
From (6), we know that the right-hand side of (A.3) is the quantity DoF bound. Therefore, dDecom ≤ dQuan . When M/N = CnA , substituting this equality into (A.2) we have dDecom = M/(K + CnA ). Considering that M/(K + A ) owing to CnA ) = N/(1 + K/CnA ) ≤ N/(1 + K/Cn−1 A Quan A = M/(K + CnA ), i.e., Cn ≥ Cn−1 , from (6) we have d dQuan = dDecom . B B Similarly, when Cn−1 < M/N ≤ CnB , we have ρ > Cn−1 Decom B ≤ and 1/ρ ≥ 1/Cn . Substituting into (A.2) we obtain d dQuan and the equality holds when M/N = CnB . Finally, we compare the proper DoF bound and the quantity A , from (6) we DoF in Region II. When CnA ≤ M/N < Cn−1 A A Quan and N ≥ (1 + K/Cn−1 )dQuan . have M ≥ (K + Cn )d A A Substituting into (5) and considering Cn + K/Cn−1 = (G − 1)K from (7a), we obtain
qnα pα n
G X
m Rj ≤ I W ; Y m j , GA + mm
(C.1a)
j=1
m Ym ≤ I W;Y m + I W ; GA |Y + mm (C.1b) j j m m m Y j ) + H(GA |Y Y j ) + mm ≤ H(Y (C.1c) m ≤ mM log γ + H(GA ) + m
(C.1d)
where m is the length of the message, Rj is the sum data rate of BS j, m diminishes when m goes to infinity, I (X; Y ) is the mutual information between X and Y , and H(X) is the entropy of X, (C.1a) follows from Fano’s inequality, (C.1b) follows from the mutual information chain rule, (C.1c) follows
13
from the entropy chain rule, and (C.1d) follows from the fact m that each BS has M antennas and GA is dependent of Y m j . By dividing log γ and m on both sides of (C.1d), and letting PG γ → ∞ and m → ∞, since limγ→∞ j=1 Rj / log γ = GKd m and limm,γ→∞ H(GA )/(m log γ) = |G A |, we obtain (17). A PPENDIX D D IMENSION OF G ENIE We prove (43a) and (43b) by mathematical induction. A A ¯ n−1 ). From (42b), we have |Gn−1 | ≤ |Gn−2 |/((G − 1)K A ¯ ¯ Since C1 = (G − 1)K from (7a) and Kn−2 = K/Kn−1 from A ¯ n−2 |G A |/C A . Therefore, (43a) (16), we obtain |Gn−1 | ≤ K n−2 1 holds for m = n − 1. Suppose that (43a) holds for m = l + 1 (l ≤ n − 2), i.e., A ¯ l |G A |/C A |Gl+1 | ≤ K l n−l−1 . Upon substituting into (42a) with m = l + 1 and after some regular manipulations, we obtain A |Gl−1 | K |GlA | ≤ ¯ · Kl (G − 1)K − C AK
(D.1)
n−l−1
¯ l−1 = K/K ¯ l from (16) and C A Considering K = n−l A (G − 1)K − K/Cn−l−1 from (7a), (D.1) becomes |GlA | ≤ ¯ l−1 |G A |/C A . Therefore, (43a) holds for m = l. K l−1 n−l As a result, (43a) is true. A A |− When m = n − 1, (43b) becomes |Gn−1 | ≤ |Sn−1 A A A A ¯ Kn−2 |Sn−1 | − |Gn−1 | /C0 . Since C0 = ∞, (43b) reduces A A to |Gn−1 | ≤ |Sn−1 |, which is exactly (42c). Therefore, (43b) holds for m = n − 1. Suppose that (43b) holds for m = l + 1 (l ≤ n − 2), i.e., A A ¯ l |S A | − |G A | /C A |Gl+1 | ≤ |Sl+1 |−K l l n−l−2 . Then we only need to prove that (43b) holds for m = l. Substituting m = A ¯ l |S A | − l + 1 into (42a) and considering |Sl+1 | = ((G − 1)K l A + A A ¯ |Sl−1 |) = (G − 1)Kl |Sl | − |Sl−1 | from (20), after some regular manipulations we obtain A A | − |Gl−1 | |Sl−1 K |GlA | ≤ |SlA | − ¯ · K Kl (G − 1)K − C A
(D.2)
n−l−2
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